On Deterministic Sketching and Streaming for Sparse Recovery and Norm Estimation

We study classic streaming and sparse recovery problems using deterministic linear sketches, including l1/l1 and linf/l1 sparse recovery problems (the latter also being known as l1-heavy hitters), norm estimation, and approximate inner product. We focus on devising a fixed matrix A in R^{m x n} and a deterministic recovery/estimation procedure which work for all possible input vectors simultaneously. Our results improve upon existing work, the following being our main contributions: * A proof that linf/l1 sparse recovery and inner product estimation are equivalent, and that incoherent matrices can be used to solve both problems. Our upper bound for the number of measurements is m=O(eps^{-2}*min{log n, (log n / log(1/eps))^2}). We can also obtain fast sketching and recovery algorithms by making use of the Fast Johnson-Lindenstrauss transform. Both our running times and number of measurements improve upon previous work. We can also obtain better error guarantees than previous work in terms of a smaller tail of the input vector. * A new lower bound for the number of linear measurements required to solve l1/l1 sparse recovery. We show Omega(k/eps^2 + klog(n/k)/eps) measurements are required to recover an x’ with |x - x’|1 <= (1+eps)|x{tail(k)}|1, where x{tail(k)} is x projected onto all but its largest k coordinates in magnitude. * A tight bound of m = Theta(eps^{-2}log(eps^2 n)) on the number of measurements required to solve deterministic norm estimation, i.e., to recover |x|_2 +/- eps|x|_1. For all the problems we study, tight bounds are already known for the randomized complexity from previous work, except in the case of l1/l1 sparse recovery, where a nearly tight bound is known. Our work thus aims to study the deterministic complexities of these problems.

💡 Research Summary

This paper investigates classic streaming and sparse recovery tasks under the stringent requirement that a single, fixed linear sketch matrix A∈ℝ^{m×n} together with a deterministic recovery/estimation algorithm Out work for all possible input vectors simultaneously. Four fundamental problems are considered: (1) ℓ∞/ℓ1 point‑query (heavy‑hitter) recovery, (2) inner‑product estimation, (3) ℓ1/ℓ1 sparse recovery with a k‑tail guarantee, and (4) ℓ2 norm estimation.

The authors first prove that the point‑query problem and inner‑product estimation are essentially equivalent: a solution to one yields a solution to the other with only a constant‑factor change in the error parameter ε. This equivalence allows the study to focus on point‑query, for which they show that any incoherent matrix—a matrix whose columns have unit ℓ2 norm and pairwise inner products bounded by ε—suffices. Incoherent matrices can be constructed via the Johnson‑Lindenstrauss (JL) lemma, almost‑k‑wise independent sample spaces, or error‑correcting codes. Using the simple deterministic recovery Out(Ax)=AᵀAx, they obtain the guarantee |x′i−x_i| ≤ ε‖x{−i}‖_1 for every coordinate i, which implies the ℓ∞/ℓ1 point‑query bound.

For measurement complexity, they prove an upper bound
 m = O(ε⁻²·min{log n, (log n / log (1/ε))²})
and cite Alon’s lower bound m = Ω(ε⁻²·log n / log (1/ε)). Thus their construction is essentially optimal up to polylogarithmic factors. By employing the Fast JL transform of Ailon and Chazelle (or later refinements), the sketch Ax can be computed in O(n·log m) time when m < n^{1/2−γ}, yielding a practical streaming algorithm.

The paper then turns to ℓ1/ℓ1 sparse recovery with a k‑tail guarantee. The best known upper bound is O(k·log(n/k)/ε²). The authors present a new lower bound
 Ω(k/ε² + k·log(εn/k)/ε)
showing that any deterministic linear sketch must use at least this many rows. The proof combines information‑theoretic arguments (to obtain the k·log(εn/k)/ε term) with a generalized Gluskin‑Kashin construction (to obtain the k/ε² term). Consequently, there is a provable separation between deterministic sketches (which must satisfy the lower bound) and randomized sketches that succeed with high probability for a fixed input (which can achieve m = O(k·log n·log³(1/ε)/√ε).

Finally, for deterministic ℓ2 norm estimation, the authors prove a tight bound
 m = Θ(ε⁻²·log(ε² n)).
The construction of A is randomized, but once fixed it works for all vectors with high probability. Recovery reduces to solving a simple convex program that approximates ‖x‖_2 within ε‖x‖_1. This matches the optimal Gelfand‑width bound and improves upon the AMS sketch, which only guarantees ε‖x‖_2 error.

Overall, the paper provides a unified framework based on incoherent matrices to address four central streaming problems under deterministic guarantees. It delivers near‑optimal measurement bounds, introduces a novel lower bound for ℓ1/ℓ1 recovery, and shows how fast JL transforms make the schemes computationally efficient. These results deepen our understanding of the deterministic complexity of linear sketches and have potential impact on applications such as distributed monitoring, network anomaly detection, and real‑time heavy‑hitter tracking where adversarial or adaptive inputs preclude reliance on randomness.